Worksheet: Fundamental Theorem of Line Integrals

In this worksheet, we will practice determining whether a vector field is conservative by finding a suitable function which produces the vector field as its gradient.

Q1:

Is there a potential for ? If so, find one.

  • A yes,
  • B no
  • C yes,
  • D yes,
  • E yes,

Q2:

Is there a potential for ? If so, find one.

  • A yes,
  • B no
  • C yes,
  • D yes,
  • E yes,

Q3:

Is there a potential for ? If so, find one.

  • A yes,
  • B no
  • C yes,
  • D yes,
  • E yes,

Q4:

Is there a potential for ? If so, find one.

  • A no
  • B yes,

Q5:

Is there a potential 𝐹 ( 𝑥 , 𝑦 ) for 𝑓 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 𝑖 𝑥 𝑦 𝑗 2 3 ? If so, find one.

  • A yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 2 𝑥 𝑦 2 2 2 3 2
  • B yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 2 + 𝑥 𝑦 2 2 2 3 2
  • C yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 2 2
  • D no
  • E yes, 𝐹 ( 𝑥 , 𝑦 ) = 𝑥 𝑦 2 2

Q6:

State whether or not the vector field 𝑓 ( 𝑥 , 𝑦 , 𝑧 ) = 𝑎 𝑖 + 𝑏 𝑗 + 𝑐 𝑘 where 𝑎 , 𝑏 , 𝑐 are constants has a potential in 3 .

  • Ayes
  • Bno

Q7:

State whether or not the vector field has a potential in .

  • Ano
  • Byes

Q8:

Is there a potential for ? If so, find one.

  • A yes,
  • B yes,
  • C yes,
  • Dno
  • E yes,

Q9:

Which of the following is true concerning the vector field defined by ?

  • A has a positive divergence at every point .
  • B , for all .
  • C is a conservative vector field.
  • D is orthogonal to the vector , for all .

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